Böhm reduction for terms and term graphs

Patrick Bahr

Invited Talk, 6th International Workshop on Confluence, 8/9/2017.

Abstract

Infinitary rewriting endows a rewriting system with a mode of
convergence that assigns an outcome to each infinite reduction that is
– in some sense – well-formed, for example, an infinite reduction
that produces the infinite list of all natural numbers. Unfortunately,
infinitary rewriting breaks well-known confluence results for lambda
calculi and orthogonal term rewriting systems. In order to recover the
confluence property in a meaningful way, Kennaway et al. extended the
ordinary reduction relation so that `meaningless terms' can be
contracted to a fresh constant ⊥. They showed that this extended
reduction relation – called Böhm reduction – enjoys the confluence
property for infinitary lambda calculi and infinitary orthogonal term
rewriting.

In this talk I give an overview of Böhm reduction and confluence in
infinitary term rewriting. The underlying mode of convergence of Böhm
reduction is based on metric spaces. As an alternative to this
approach, I present a mode of convergence – based on partially
ordered sets – that enjoys the confluence property for orthogonal
term rewriting systems directly, i.e. without the need to explicitly
contract meaningless terms. Finally, I sketch how a corresponding
infinitary rewriting theory can be developed for term graphs.